Exploration of the Theoretical and Empirical Relationships Between Entropy and Diffusion

Abstract: Knowledge and control of chemical engineering systems requires acquiring worths for process variables and functions that vary in difficulty of computation and measurement. Today report aimed to show the connections in between entropy and diffusion and to highlight the avenues to convert data from one into the other. The correlation in between the two […]



Abstract:

Knowledge and control of chemical engineering systems requires acquiring worths for process variables and functions that vary in difficulty of computation and measurement. Today report aimed to show the connections in between entropy and diffusion and to highlight the avenues to convert data from one into the other. The correlation in between the two principles was explored at the microscopic and single-particle level. The scope of expedition was limited to the particle level in order to determine commonness that underlie higher-level phenomena. A probabilistic model for molecular diffusion was established and provided to illustrate the close coupling between entropic information and diffusion. The relationship in between diffusivity and configurational/excess entropy was expounded by examining the Adam-Gibbs and Rosenfeld relations. A modified analog of the Adam-Gibbs relation was then found to accurately forecast experimental data on diffusion and translational entropy of single water particles. The quantitative relations stated in this report allow the chemical engineer to get info on the abstract entropy capacity by mapping from more concrete dynamical homes such as the diffusion coefficient. This correspondence fosters greater insight into the operations of chemical engineering systems granting the engineer increased opportunity for control while doing so.

Introduction:

Systems, whether observed or simulated, include the complex interaction between numerous degrees of liberty, both of time and area. The analysis of chemical engineering systems, in specific, regularly needs understanding of both thermodynamic capacities and dynamic state variables. The set of thermodynamic capacities that appear in the analysis of these systems consist of enthalpy, entropy and complimentary energy as members. Each of these potentials is a function of system variables such as pressure, temperature and structure. This dependence on the system’s specifications allows the thermodynamic potentials, together with their very first and 2nd derivatives, to constrain the stability and stability of chemical systems. The constraining ability of these potentials derives from the first and 2nd law of thermodynamics, entropy maximization concepts and arguments from mathematical analysis.

Occupation of states of equilibrium and stability is only one element of a system; it is likewise crucial to understand how systems progress towards or far from these states. Dynamic processes, such as transport phenomena, mediate this time development. Transportation phenomena incorporate the motion of conserved quantities: heat, mass and momentum. The motion of mass, heat and momentum represent the pathways systems trace out in state space. The full description, understanding and control over chemical engineering systems require understanding of the active dynamic and thermodynamic procedures, and their correlations, of the system.

This report will focus on the relationship in between entropy and diffusion. Diffusion signifies a process that systems undergo in action to some non-uniformity or asymmetry in the system. Entropy generation can be understood as an effect of diffusional phenomena. It is the apparent interconnection between the two principles that this report plans to highlight and characterize. This report intends to define relations between entropy and diffusion so that it is possible to translate qualitative and quantitative information between the two.

Theory and Procedure:

Entropy ( S) is recognized as a measure of the size of setup area where setup area is the area of all possible microscopic configurations a system can inhabit with a specific likelihood. This is stated with Gibbs entropy formula,
S=- k_b ∑ & sum; p_i lnâ ¡ (p_i) ≡, k_b ≡ & equiv; Boltzmann continuous, p_i & equiv; likelihood of microstate.
If the likelihood of each microstate is equal then,Ω
Ω ≡ S= k_b ln & Omega;-LRB- , where & Omega;-LRB- & equiv; variety of tiny setups consistent with equilibrium state. These expressions for thermodynamic entropy closely resemble the expression for info theoretic entropy and indicate that entropy can be viewed as a measure of the degree of unpredictability about a system triggered by details not being interacted by macrostate variables, like pressure and temperature, alone. Tiny configurations are determined by the vibrational, rotational and translational degrees of flexibility of the molecular constituents of a system. As such, any process that increases the number of tiny setups available to a system will also increase the degree of the system’s configuration area, consequently, raising its entropy.

Diffusion is defined as a process where a types moves from a region of high chemical capacity to an area of low chemical capacity; without loss of generality, the driving force for particle movement is regularly a concentration difference. This is recorded with Fick’s First Law of Diffusion, J = -D∇& nabla; c ∇ )with & nabla;=( d/dx, d/dy, d/dz) ≡ , where J & equiv; diffusive flux, c ≡ & equiv; concentration,D& equiv; diffusion coefficient. Fick’s Second Law asserts the time reliance of a concentration∇profile,
& part; c/ & part; t =& nabla; ∠™ D & nabla; c From the above equations, diffusion can be conceived as a response function, whose value is figured out by a forcing function( gradient in concentration ), which seeks to decrease the forcing function to absolutely no. The translational motion of the particles will continue up until a state of consistent particle distribution is accomplished. Equivalently, diffusion is the process by which a system shifts from a non-equilibrium configuration towards one that more carefully resembles a stability state, that being, a state where the chemical potentials of all types are equivalent.

Although primary, the theoretical information presented above identifies a unifying link between the 2 ideas, stage area growth. Entropy is the control variable for this growth whereas diffusion is the procedure. This connection will be displayed by first presenting and relating probability based descriptions of particle diffusion and entropy. By evaluating the relationship between the diffusion coefficient and entropy terms, a more extension of the linkage in between the 2 will be arrived at. A focus on single water molecules will even more illustrate and support the connection between diffusion and entropy.

Outcomes and Discussion:

The molecular movements executed by particles were exposed to be reducible to a probabilistic design incorporating statistical mechanical arguments in Albert Einstein’s 1905 Investigation on the Theory of Brownian Motion (14-18). The presumption that each particle underwent motion, limited to the single x co-ordinate, independently of neighboring particles was advanced; this was achieved by picking time periods of movement (τ& tau;-LRB- ) and area (Δ& Delta; x) to not be too little. A particle density function f( x, t) which reveal the variety of particles per system volume was presumed. This likelihood density function was formed by the spatial increments particles traveled over the time period. This function was then broadened in a Taylor series yielding,
f( x ∠† x, t)= f( x, t) ∠† ∂& part; f (∂x, t)/ & part; x ∠† ^ 2/2!( & part; ^ 2 f (x, t))/( & part; x ^ 2) ∠™ ∠™ ∠™ ad infΔ.
f( x, t & tau;) dx= dx & int; _( ∠†= m) ^ (∠†Δ= & infin;) f( x ∠† )Ï & bull;-LRB- & Delta;) d & Delta;-LRB-
∂This expansion can be incorporated, considering that only little values of & Delta;-LRB- add to the function.
f ∫ & part; f/ & part; t∠™ & tau;= f & int; _(-∞& infin;-RRB- ^ & infin;-LRB- Ï & bull;-LRB- ∠† )d∠† & part; x/ & part; f & int; _ (- & infin;-RRB- ^ & infin;-LRB- ∠† Ï & bull;-LRB- ∠†)) d∠† (& part; ^ 2 y )/ (& part; x ^ 2) & int; _(- & infin;-RRB- ^ & infin;-LRB- ∠† ^ 2/2 )Ï & bull;-LRB- ∠†) d∠† ∠™ ∠™ ∠™
The first essential on the right-hand side is unity by the procedure of a probability space whereas the second and other even terms disappear due to space proportion (∂ ∫Ï & bull;( x) =Ï & bull;-LRB– x)
∞What stays after this simplification is
& part; f/ & part; t=( & part; ^ 2 f)/( & part; x ^ 2) & int; _ (- & infin;-RRB- ^ & infin;-LRB- ∠† ^ 2/2 & tau;-RRB- Ï & bull;-LRB- ∠†∂) d∠† & int; _(- & infin;-RRB- ^ & infin;-LRB- Ï & bull;-LRB- ∠†) )d∠†
whereby setting the term after the 2nd derivative toD leads to & part; f/√& part; t= D√( & part; ^ 2 f)/( & part; x ^ 2) which is Fick’s Second Law. Fixing the above important equation generates the particle density function,
f (x, t)= n/ & radic; 4 & pi;D e ^(- x ^ 2/4Dt)/ & radic; t
This is a typical distribution that has√the distinct residential or commercial property of having the maximum entropy of any other continuous circulation for a specified mean and variation, equivalent to 0 and & radic; 2Dt , respectively, for the particle distribution above. Einstein later found that the mean displacement( diffusion) of particles & lambda; x which depends upon temperature, pressure, Avogadro’s number N and the Boltzmann constant k_b to be,
& lambda; _ x= & radic; t∠™ & radic;-LRB-( RT & int; _(- & infin;-RRB- ^ & infin;-LRB- Ï & bull;-LRB- ∠†)) d∠†)/( 3 & pi; kPN)
It is fascinating that measurable physical homes such as the diffusion coefficient appear in a mathematical model that makes sure maximization of entropy.

Equation-based relationships between diffusion and entropy have been investigated for many years. One such relation is,
D( T) = D( T= T_0) e ^( C/( TS_c )),
where S_c the configuration entropy of the system specified as,
S_c (T) = S( T)- S_vib( T)
and S_vib is the vibrational entropy of the system and D( T_0) is the diffusion coefficient at some higher temperature level T_0 This is referred to as the Adam-Gibbs relation and explicates the strong reliance diffusion has on entropy. The Rosenfeld relation in between the diffusion coefficient and entropy provides another fascinating connection,
D = a∠™ e ^((( bS_ex)/ k_b ))
S_ex
is excess entropy found by deducting the entropy of an ideal gas at the very same conditions from the system’s total entropy, a and b serve as fitting parameters and k_b is the Boltzmann’s continuous. These above expressions broadcast a noticable and well-founded connection in between diffusion and entropy to the degree that understanding one allows the decision of the other.

Saha and Mukherjee in their post “Connecting diffusion and entropy of bulk water at the single particle level,” executed molecular vibrant simulations to establish a linkage between thermodynamic and vibrant residential or commercial properties of private water particles (825-832). Translational ( S_trans) and rotational ( S_rot) entropies were calculated at varying temperatures in addition to computations of self-diffusion coefficient ( D) consequently allowing the building of a generalization of the Adam-Gibbs relation above to relate configurational entropy with translation relaxation (self-diffusion) time. S_trans was examined from the entropy of a solid-state quantum harmonic oscillator as revealed below,
S_trans ^ QH = k_b ∑& amount; _( i=⁄1) ^ 3(( â „ ω& omega; _ i) & frasl;( k_b T))/ e ^(( â „ & omega; _ i) & frasl; (k_b T) )- lnâ ¡ (1-e ^( (â „ & omega; _ i) & frasl; (k_b T)) )
where T suggests temperature level,k_b is the Boltzmann consistent and â „= h/2 & pi;-LRB- , hbeing the Planck constant. A technique known as permutation decrease which thinks about water molecules to be indistinguishable and to reside in an effective localized setup area was made use of to get a covariance matrix of translational variations of each permuted particle along the x, y and z co-ordinates. This produced a 3×3 matrix, whereupon diagonalization of the matrix produced 3 eigenvalues and 3 frequencies (ω& omega; i), which were input to the expression above. Diffusion was evaluated with the Vogel-Fulcher-Tammann (VFT) formula,
D ^( -1) (T) = D_0 ^( -1) e ^[1/(K_VFT (T/T_VFT -1))]
with KVFT denoting the kinetic fragility marker and TVFT symbolizing the temperature at which the diffusion coefficient diverges. The idea of thermodynamic fragility, which appears in the above analysis, measures the rate at which dynamical homes such as inverse diffusivity grow with temperature level. According to IUPAC Compendium of Chemical Terminology, self-diffusion is the diffusion coefficient ( D_i *) of types i when the chemical potential gradient is absolutely no ( a is the activity coefficient and c is the concentration).
D_i = D_i (∂& part; lnc_i )/ (& part; lna_i)
Saha and Mukherjee fitted the variant of the Adam-Gibbs formula D= ae ^(( bS_trans⁄& frasl; k_b)) to their information.

The Pearson’s correlation coefficient (R), which is the covariance of two variables divided by the item of their basic discrepancies, attained a worth of 0.98 This worth suggests a directed and strong statistical association between translational entropy and translational diffusivity. Such a great fit indicates that an underlying physical relation in between entropy and diffusion does exist which one can convert understanding of characteristics, information that requires fewer computational resources, to an understanding of thermodynamics, information that is computationally more expensive. As communicated by the authors, this connection was confirmed for a specific system and generalization of its findings to other systems ought to take place only upon application of the very same methods to other systems. Nonetheless, if extra analysis can provably please empirical and theoretical restrictions, the approaches detailed above can supply insight to more complicated environments.

Conclusion:

Controllability, a notion open up to a number of meanings, can be considered the capability to move a system between different areas of its setup area through the application of a particular number of permissible manipulations. The supreme objective of chemical engineering analysis is the capability to identify the output of some system through the rational and systematic control of input variables. This controllability enables optimization of processes such as separations. Nevertheless, without the ability to keep an eye on a systems response to perturbations, it becomes challenging to know in what instructions or to what degree a modification ought to be performed. Hence, controllability indicates observability of procedure variables; or state differently, all pertinent procedure variables can be determined to some level.

This report focused specifically on the interconnection between diffusion and entropy. Both of these entities are necessary in the design, characterization and control of engineering systems. A barrier to achieve full control develops from the problem of attaining and determining abstract quantities such as entropy. An approach to overcome this challenge is to identify a one-to-one correspondence between the intractable variable and one that is more certified and more easily determined. Diffusion and the related diffusion coefficient represent the home that abides by computational and empirical methods and enables conclusion of the mapping. The equations and relations provided above are structurally varied and apply to various conditions however show that from knowledge of a system’s characteristics (diffusivity) one obtains understanding of the system’s thermodynamics.

Recommendations:

Engel, Thomas and Philip Reid. Physical Chemistry. San Francisco: Pearson Benjamin Cummings,2006
Seader, J.D, Ernest J. Henley and D. Keith Roper. Separation Process Concepts: Chemical and Biochemical Operation 3rd Edition. New Jersey: John Wiley & Sons, Inc., 2011
Einstein, Albert. “Investigation on The Theory of Brownian Motion.” ed. R. Furth. Trans. A. D. Cowper. Dover Publications, 1926 and1956
Seki, Kazuhiko and Biman Bagchi. “Relationship between Entropy and Diffusion: An analytical mechanical derivation of Rosenfeld expression for a rugged energy landscape.” J. Chem. Phys. 143(19),2015 doi: 10.1063/ 1.4935969
Rosenfeld, Yaakov. “Relation between the transport coefficients and the internal entropy of easy systems,” Phys. Rev. A 15, 2545, 1977
Rosenfeld, Yaakov. “A quasi-universal scaling law for atomic transportation in easy fluids.” J. Phys.: Condensed Matter 11, 5415,1999
Sharma, Ruchi, S. N. Chakraborty and C. Chakravarty. “Entropy, diffusivity, and structural order in liquids with waterlike abnormalities.” J. Chem. Phys. 125,2006 Doi: 10.1063/ 1.2390710
Saha, Debasis and Arnab Mukherjee. “Linking diffusion and entropy of bulk water at the single particle level.” J. Chem. Sci. 129( 7 ),2017 Pg 825-832 Doi: 10.1007/ s23039-017-1317- z
Hogg, Robert V. and Elliot A. Tanis. Probability and Analytical Inference 6th Edition. Prentice-Hall Inc., 2001

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